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7. Let D = {z C z 1) denote the closed unit disc centered at the origin. Let f : D → C be a conti...
12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3.
12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3.
Let Ω be an open set and a E Ω with (the closed disc) D(a,p) Ω Let f є H(Q). We have proved that for any r 〈 ρ, f has a power series expansion in the open disc D(a,r) CO 0 where, for all n0,1,2 7l Here C is the positively oriented circle: z-a+pe.θ, 0-θ-2π. In particular, f has a Taylor series expansion in D(a, r): f" (a) 2-a 0 This results in two consequences (will be shown in...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
4. Suppose that f is an integrable function on R. Let denote the centered Hardy-Littlewood maximal function. (i) Show that M/is lower semicontinuous Recall: A real-valued function g is lower semicontinuous if and only if the superlevel se t(g > λ} is open for all λ E R.
4. Suppose that f is an integrable function on R. Let denote the centered Hardy-Littlewood maximal function. (i) Show that M/is lower semicontinuous Recall: A real-valued function g is lower semicontinuous if...
(7) Let V be the region in R3 enclosed by the surfaces+2 20 and z1. Let S denote the closed surface of V and let n denote the outward unit normal. Calculate the flux of the vector field F(x, y, z) = yi + (r2-zjy + ~2k out of V and verify Gauss Divergence Theorem holds for this case. That is, calculate the flux directly as a surface integral and show it gives the same answer as the triple integral...
c. Evaluate ,f(z) dz with า the circle of radius 1 centered at the origin and traveled once counterclockwise ˊ们: (1-2 For real twith-1 < t < 1 and +12)-1 Explain why f(:)) has an expansion of the form in C , let f(z) be defined by fG)- a. b. Compute Uo(t), Ui(t), and Uz(t) in terms of t. c. Recalling that t is a real number smaller than 1 in absolute value, find the radius of convergence of this...
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
1. Let C denote the closed curve of intersection of the hemisphere z = (25 – x’ - y²); and the cylinder x² + y? = 9 oriented counterclockwise. Compute fF.ds where F =(-y,x,z).
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P
rty. I 5. [16...